19 Elliptic IntegralsSymmetric Integrals19.23 Integral Representations19.25 Relations to Other Functions

The condition $y\le z$ for (19.24.1) and (19.24.2) serves only to identify $y$ as the smaller of the two nonzero variables of a symmetric function; it does not restrict validity.

19.24.1 | $$\mathrm{ln}4\le \sqrt{z}{R}_{F}(0,y,z)+\mathrm{ln}\sqrt{y/z}\le \frac{1}{2}\pi ,$$ | ||

$$, | |||

19.24.2 | $$\frac{1}{2}\le {z}^{-1/2}{R}_{G}(0,y,z)\le \frac{1}{4}\pi ,$$ | ||

$0\le y\le z$, | |||

19.24.3 | $${\left(\frac{{y}^{3/2}+{z}^{3/2}}{2}\right)}^{2/3}\le \frac{4}{\pi}{R}_{G}(0,{y}^{2},{z}^{2})\le {\left(\frac{{y}^{2}+{z}^{2}}{2}\right)}^{1/2},$$ | ||

$y>0$, $z>0$. | |||

If $y$, $z$, and $p$ are positive, then

19.24.4 | $$\frac{2}{\sqrt{p}}{(2yz+yp+zp)}^{-1/2}\le \frac{4}{3\pi}{R}_{J}(0,y,z,p)\le {(yz{p}^{2})}^{-3/8}.$$ | ||

Inequalities for ${R}_{D}(0,y,z)$ are included as the case $p=z$.

A series of successively sharper inequalities is obtained from the AGM process (§19.8(i)) with ${a}_{0}\ge {g}_{0}>0$:

19.24.5 | $$\frac{1}{{a}_{n}}\le \frac{2}{\pi}{R}_{F}(0,{a}_{0}^{2},{g}_{0}^{2})\le \frac{1}{{g}_{n}},$$ | ||

$n=0,1,2,\mathrm{\dots}$, | |||

where

19.24.6 | ${a}_{n+1}$ | $=({a}_{n}+{g}_{n})/2,$ | ||

${g}_{n+1}$ | $=\sqrt{{a}_{n}{g}_{n}}.$ | |||

Other inequalities can be obtained by applying Carlson (1966, Theorems 2 and 3) to (19.16.20)–(19.16.23). Approximations and one-sided inequalities for ${R}_{G}(0,y,z)$ follow from those given in §19.9(i) for the length $L(a,b)$ of an ellipse with semiaxes $a$ and $b$, since

19.24.7 | $$L(a,b)=8{R}_{G}(0,{a}^{2},{b}^{2}).$$ | ||

For $x>0$, $y>0$, and $x\ne y$, the complete cases of ${R}_{F}$ and ${R}_{G}$ satisfy

19.24.8 | ${R}_{F}(x,y,0){R}_{G}(x,y,0)$ | $>\frac{1}{8}{\pi}^{2},$ | ||

${R}_{F}(x,y,0)+2{R}_{G}(x,y,0)$ | $>\pi .$ | |||

Also, with the notation of (19.24.6),

19.24.9 | $$\frac{1}{2}{g}_{1}^{2}\le \frac{{R}_{G}({a}_{0}^{2},{g}_{0}^{2},0)}{{R}_{F}({a}_{0}^{2},{g}_{0}^{2},0)}\le \frac{1}{2}{a}_{1}^{2},$$ | ||

with equality iff ${a}_{0}={g}_{0}$.

Inequalities for ${R}_{-a}(\mathbf{b};\mathbf{z})$ in Carlson (1966, Theorems 2 and 3) can be applied to (19.16.14)–(19.16.17). All variables are positive, and equality occurs iff all variables are equal.

19.24.10 | $$\frac{3}{\sqrt{x}+\sqrt{y}+\sqrt{z}}\le {R}_{F}(x,y,z)\le \frac{1}{{(xyz)}^{1/6}},$$ | ||

19.24.11 | $${\left(\frac{5}{\sqrt{x}+\sqrt{y}+\sqrt{z}+2\sqrt{p}}\right)}^{3}\le {R}_{J}(x,y,z,p)\le {(xyz{p}^{2})}^{-3/10},$$ | ||

19.24.12 | $$\frac{1}{3}(\sqrt{x}+\sqrt{y}+\sqrt{z})\le {R}_{G}(x,y,z)\le \mathrm{min}(\sqrt{\frac{x+y+z}{3}},\frac{{x}^{2}+{y}^{2}+{z}^{2}}{3\sqrt{xyz}}).$$ | ||

Inequalities for ${R}_{C}(x,y)$ and ${R}_{D}(x,y,z)$ are included as special cases (see (19.16.6) and (19.16.5)).

Other inequalities for ${R}_{F}(x,y,z)$ are given in Carlson (1970).

If $a$ ($\ne 0$) is real, all components of $\mathbf{b}$ and $\mathbf{z}$ are positive, and the components of $z$ are not all equal, then

19.24.13 | ${R}_{a}(\mathbf{b};\mathbf{z}){R}_{-a}(\mathbf{b};\mathbf{z})$ | $>1,$ | ||

${R}_{a}(\mathbf{b};\mathbf{z})+{R}_{-a}(\mathbf{b};\mathbf{z})$ | $>2;$ | |||

see Neuman (2003, (2.13)). Special cases with $a=\pm \frac{1}{2}$ are (19.24.8) (because of (19.16.20), (19.16.23)), and

19.24.14 | ${R}_{F}(x,y,z){R}_{G}(x,y,z)$ | $>1,$ | ||

${R}_{F}(x,y,z)+{R}_{G}(x,y,z)$ | $>2.$ | |||

The same reference also gives upper and lower bounds for symmetric integrals in terms of their elementary degenerate cases. These bounds include a sharper but more complicated lower bound than that supplied in the next result:

19.24.15 | $${R}_{C}(x,\frac{1}{2}(y+z))\le {R}_{F}(x,y,z)\le {R}_{C}(x,\sqrt{yz}),$$ | ||

$x\ge 0$, | |||

with equality iff $y=z$.